Chapter 9 Error-Correcting Constrained Coding

Chapter

ErrorCorrecting Constrained Co ding

In this chapter we consider co des that have combined errorcorrection and constrained

prop erties We b egin with a discussion of error mechanisms in recording systems and the

corresp onding error typ es observed We then discuss asso ciated metrics imp osed on con

strained systemsprimarily the Hamming Lee and Euclidean metricsand we survey the

literature on b ounds and co de constructions In addition we consider two imp ortant classes

of combined errorcorrectionconstrained co des sp ectral null co des and forbidden list co des

Errormechanisms in recording channels

Magnetic recording systems using p eak detection as describ ed in Chapter of this chapter

are sub ject to three predominant typ es of errors at the peak detector output The most

frequently observed error is referred to as a bitshift error where a pair of recorded symbols

is detected as a left bitshift or the pair is detected as a right bitshift Another

commonly o ccurring error typ e is called a dropout error or sometimes a missingbit error

where a recorded symbol is detected as a Less frequently a dropin error or extrabit

error results in the detection of a recorded as a It is convenient to refer to the dropin

and dropout errors as substitution errors

Hammingmetric constrained co des are most p ertinent in recording channels that b ehave

which dropin and dropout errors o ccur with equal like a binary symmetric channel in

probability However there are alternative mo dels of interest that suggest the use of co des

designed with other criteria in mind beyond optimization of minimum Hamming distance

Among these mo dels the two that have received the most attention are the asymmetric

channelwhere only dropin errors or dropout errors but not both are encountered and

the bitshift channelwhere a symbol is allowed to shift p osition by up to a presp ecied

umber of p ositions n

CHAPTER ERRORCORRECTING CONSTRAINED CODING

Another error typ e we will consider is a synchronization error resulting in an insertion

or deletion of a symbol in the detected symbol sequence In practical digital recording

systems on disks and tap e this typ e of error can have catastrophic consequences with re

gard to recovery of information that follows the synchronization loss As a result recording

devices use synchronization and clo ck generation techniques in conjunction with co de con

the G constraint in straints such as the k constraint in RLL co des for p eak detection and

PRML GI co des to eectively preclude such errors Nevertheless RLLconstrained

synchronizationerrorcorrecting co des have some intrinsic co dingtheoretic interest and we

will discuss them b elow Co des capable of correcting more than one insertion and deletion

error may also b e used to protect against bitshift errors which result from the insertion and

deletion of s on either side of a The edit distance or Levenshtein metric and the Lee

metric arise naturally in the context of synchronization errors

In recording systems using partialresp onse with some form of sequence detection ex

emplied by the PRML system describ ed in Chapter the maximumlikelihood detector

tends to generate burst errors whose sp ecic characteristics can b e determined from the error

events asso ciated with the underlying trellis structure We will briey survey various trellis

co ded mo dulation approaches for PRML that yield co des whichcombine GI constraints

with enhanced minimum Euclidean distance

In practice constrained co des must limit error propagation Slidingblo ck deco ders of the

most frequently used d k RLL co des and PRML GI co des typically will propagate

a single detector error into a burst of length no more than eight bits For example the

maximum error propagation of the industry standard RLL and RLL co des are

four bits and ve bits resp ectively and the PRML co de limits errors to a single byte

The conv entional practice in digital recording devices is to detect and correct such errors by

use of an outer errorcorrecting co de such as a Fire co de interleaved ReedSolomon co de

or a mo dication of such acode

Gilb ertVarshamovtyp e lower b ounds

Classical bound for the Hamming metric

There are several error metrics that arise in the context of digital recording using constrained

sequences For substitutiontyp e errors and bitshift errors p ossibly propagated into burst

errors by the mo dulation deco der it is natural to consider errorcorrecting co des based up on

the Hamming metric It is therefore of interest to investigate Hamming distance prop erties

of constrained sequences

The Gilb ertVarshamov b ound provides for unconstrained sequences over a nite alphab et

alower b ound on the size of co des with presp ecied minimum Hamming distance In this

CHAPTER ERRORCORRECTING CONSTRAINED CODING

section we present bounds of the Gilb ertVarshamov typ e and apply them to the class of

runlengthlimited binary sequences

Let denote a nite alphab et of size jj and denote the Hamming distance b etween two

q q

w r be the Hamming words w w by w w For a word w let B

Hamming

sphere of radius r in centered at w that is

w w r g w r fw B

Hamming

q q

Let V w r be the cardinality or volume of the sphere B w r This quantity is

jj indep endent of the center word w so we will use the shorthand nota

tion V r

The Gilb ertVarshamov b ound provides a lower bound on the achievable cardinalit y M

of a subset of with minimum Hamming distance at least d We will refer to such a subset

as a Mdco de

Theorem There exists a Mdcode with

V d

For future reference we recall that the pro of of this b ound is obtained by iteratively

selecting the l th co deword w in the co de from the complement of the union of Hamming

spheres of radius q centered at the previously selected co dewords B w d

q q

Continuing this pro cedure until the union of spheres exhausts an Mdco de is

obtained whose size M satises the claimed inequality

Let dq denote the relative minimum distance and let H z log

log logz for z be a z ary generalization of the entropy

function The Gilb ertVarshamov b ound in terms of the rate R of the resulting co de can b e

expressed as

log M log V d

R log jj log jjH jj

q q

for the last inequalitywe refer the reader to Berl pp

Hammingmetric b ound for constrained systems

Any generalization of the Gilb ertVarshamov bound to a constrained system S must take

into account that the volumes of Hamming spheres in S are not necessarily indep endent

of the sp ecied centers Before deriving such b ounds we require a few more denitions Let

CHAPTER ERRORCORRECTING CONSTRAINED CODING

X denote an arbitrary subset of For a word w X we dene the Hamming sphere of

radius r in X by

B w r B w r X

The maximum volume of the spheres of radius r in X is

V r maxjB w r j

Xmax X

w X

and the average volume of spheres of radius r in X is given by

r V jB w r j

X X

jX j

w X

We also dene the set B r of pairs w w ofwords in X at distance no greater than r

B r fw w w w r g

X Hamming

Note that jB r j V r jX j Finallywe dene an X M dco de to b e a Mdco de

X X

that is a subset of X

tforward application of the Gilb ertVarshamov construction yields the following A straigh

result

Lemma Let X be a subset of and d be a positive integer Then there exists an

X M dcode with

jX j

V d

Xmax

The following generalization of the Gilb ertVarshamov bound rst proved by Kolesnik

and Krachkovsky KolK is the basis for the more rened bounds derived later in the

section It provides a b ound based up on the average volume of spheres rather than the

maximum volume as was used in Lemma

Lemma Let X be a subset of and d be a positive integer Then there exists an

ode with X M dc

jX j jX j

V d jB d j

X X

Pro of Consider the subset X of words w X whose Hamming spheres of radius d

satisfy jB w dj V d The subset X must then satisfy jX j jX j If we

X X

iteratively select co dewords from X following the pro cedure used in the derivation of the

Gilb ertVarshamov b ound we obtainanX M dco de where

jX j

j jX jX j

V d V d jB d j

X X X

CHAPTER ERRORCORRECTING CONSTRAINED CODING

as desired

In general neither of the b ounds in the preceding two lemmas is strictly sup erior to the

other as observed byGuandFuja GuF However using an analysis of a new co de search

algorithmdubb ed the altruistic algorithm to distinguish it from the greedy algorithm

that lies at the heart of the standard Gilb ertVarshamov form of b oundthey eliminated

the factor of in the denominator of the b ound in Lemma This improved lower b ound

stated below as Lemma is always at least as go o d as the b ound in Lemma and a

strict improvement over Lemma

The key element of the improved co de search algorithm is that at eachcodeword selection

step the remaining p otential co deword with the largest number of remaining neighb ors at

distance dorlesstakes itself out of consideration As noted in GuF a similar approach

was develop ed indep endently by Ytrehus Yta who applied it to compute b ounds for

runlengthlimited co des with various error detection and correction capabilities Ytb

Lemma Let X be a subset of and d be a positive integer Then there exists an

X M dcode with

jX j jX j

V d jB dj

X X

Kolesnik and Krachkovsky KolK applied Lemma to sets X consisting of words

of length q in runlengthlimited and charge constrained systems Their asymptotic lower

bound was based up on an estimate of the average volume of constrained Hamming spheres

whose centers ranged over all of S Their estimate made use of a generating function

for pairwise q blo ck distances in these families of constrained systems

Improved Hammingmetric b ounds

Marcus and Roth MR found improved b ounds by considering subsets X of S where

dep ending up on the designed relative minimum distance are im additional constraints

posed up on the frequency of o ccurrence of co de symbols w We now discuss the

derivation of these b ounds

Let S be a constrained system over presented by an irreducible deterministic graph

G V E L Denote byG the set of all stationary Markovchains on G see Section

The entropy of P Gis denoted by HP

Given a stationary Markovchain P G along with a vector of realvalued functions

f f f f E IR wedenoteby E f the exp ected value of f with resp ect to P

t G P

E f P ef e

eE G

CHAPTER ERRORCORRECTING CONSTRAINED CODING

For a subset W fw w w g of we dene the vector indicator function I

t W

E IR by I I I I where I E IR is the indicator function for a

G W w w w w G

symbol w

if L ew

I e

otherwise

Let G G denote the lab eled pro duct graph dened by V V V fu u

GG G G

u u V g and E E E There is an edge e e in G G from state u u to

G GG G G

state v v whenever e is an edge in G from state u to state u and e is an edge in G from

e e L eL e We dene state v to v The lab eling on G G is dened by L

G G G G

on E the co ordinate indicator functions I and I given by I e e I e and

GG W

W W W

I e e I e Finallywe dene the co ordinate distance function D E IRby

W GG

if L e L e

G G

D e e

otherwise

For agiv en symb ol subset W of size t and a vector p we now dene the quantities

S p sup HP

P G

E I p

P W

and

T p sup HP

P G G

E I p i

D E

Finally to concisely state the b ounds weintro duce the following function of the relative

designed distance

R sup f S p T p g

W W W

The following theorem is proved in MR It is obtained by application of Lemma

to the words in S generated by cycles in G starting and ending at a sp ecied state

u G with frequency of o ccurrence of the symbols w W given approximately by p for

i i

i t

Let S beaconstrained system over presented by a primitive determinis Theorem

tic graph let and let W beasubsetof of size t Then thereexist S Mqcodes

satisfying

log M

R o

where o stands for a term that goes to zero as q tends to innity

CHAPTER ERRORCORRECTING CONSTRAINED CODING

Computation of the quantities S pandT p requires the solution of a constrained

W W

optimization problem in which the ob jective function P HP is concave and the con

straints are linear The theory of convex duality based up on Lagrange multipliers provides a

metho d to translate the problem into an unconstrained optimization with a convex ob jective

function MR

In order to reformulate the problem weneedtointro duce a vectorvalued matrix function

t t

that generalizes the adjacency matrix A For a function f E IR and x IR let

G G

A x be the matrix dened b y

xf e

A x

e eu ev

We remark that for any function f the matrix A is precisely the adjacency matrix of

The following lemma is the main to ol in translating the constrained optimization problem

to a more tractable form It is a consequence of standard results in the theory of convex

duality

Lemma Let G and f be as above Let g E IR and dene f g E

G G

tl t l

IR Then for any r IR and s IR

n o

sup x r z s logA x z HP inf

x IR

P G

z IR

E f r

E g s

can derive dual formulas for the lower b ounds Applying Lemma to Theorem we

R for a sp ecied symbol set W and relative minimum distance For the case where

W consists of a single symbol w the resulting formula is particularly tractable To

express it succinctly we dene

h i

J I I D E IR

w GG

fw g fw g

and to simplify notation S pS p T p T p andR R

w w w

fw g fw g fw g

The lower bound on attainable rates follows from the following theorem

Theorem Let S be a constrained system over presented by a primitive graph G

let and let w Then

S p inf fpx log A xg

w GI

xIR

CHAPTER ERRORCORRECTING CONSTRAINED CODING

T p inf fpx z log A x z g

w GGJ

xIRz IR

n o

R sup inf px logA x

w GJ

xIR

o n

x z inf px z logA

J GG

xIRzIR

In particular if P Ghas maximal entropy rate

HP sup HP

P G

and the symb ol probability p equals E I then

P w

S p logA

w G

and setting x in part b of Theorem

T p inf fz logA zg

w GGJ

z IR

From Theorem and part c of Theorem we recover the lower b ound of Kolesnik

and Krachkovsky

Corollary

log M

R o

where

R logA inf fz logA zg

KK G GGJ

z IR

Better lower b ounds can b e obtained by prescribing the frequency of o ccurrence of words

w of arbitrary length rather than only symb ols See MR for more details

Example For the RLL constrained system consider the cases where W

fg and W fg with corresp onding lower bounds R and R It is not dicult to

see that R must equal R Table from MR gives the values of R R

and R for selected values of

We remark that in some circumstances one might assign to each edge e E a cost

asso ciated to its use in a path generating a sequence in S Lower bounds on the rate of

co des into S with sp ecied relative minimum distance and average cost constraint have

been derived by Winick and Yang WY and Khayrallah and Neuho KN

CHAPTER ERRORCORRECTING CONSTRAINED CODING

R R R

Table Attainable rates for RLL constrained system

Towards spherepacking upp er b ounds

In comparison to lower b ounds much less is known ab out upp er b ounds on the size of

blo ck co des for constrained systems We describ e here a general technique intro duced by

Ab delGhaar and Web er in AW Let S be a constrained system over and let X be

q q

a nonempty subset of For a word w denote by B w t the set of words w X

which are at distance t or less from w according to some distance measure w w If C is

an S Md t co de then by the spherepacking bound we must have

jB w tj jX j

w C

for any nonempty subset X In the conventional spherepacking bound the subset X

ed bounds may be obtained by taking X to be a is taken to be the whole set Improv

prop er subset of Sp ecically dene

N S X ijfw S jB w tj igj

w t then B Now if X is contained in

w S

jX j

iN S X i jX j

so there exists an integer j j jX j suc h that

iN S X i jX j

and

iN S X i jX j

CHAPTER ERRORCORRECTING CONSTRAINED CODING

Ab delGhaar and Web er AW used these inequalities to establish the following upp er

bound on the co de cardinality

Theorem Let C be an S Md t code and let X be a nonempty subset

B w t Then of

w S

jX j iN S X i

jC j N S X i

Pro of If jC j N S X i then we are done already So wemay assume that jC j

P P

j j

N S X i Divide C into two subsets C C where C consists of the N S X i

i i

elements w of C with the smallest jB w tj Then

X X

iN S X i jB w tj

w C

and

X X

jB w tj j jC j N S X i

w C

Now use the preceding two inequalities to lower b ound the lefthand side of inequality

j j

X X X

iN S X ij jC j N S X i jB w tj jX j

i i

w C

The theorem follows from this

For the sp ecial case of bitshift errors Ab delGhaar and Web er obtain in AW upp er

b ounds on singlebitshift correcting co des C for d k RLL constrained systems S as follows

First partition every co de C S into constantweight subsets C C such that

w w

each element of C has Hamming weight w then apply Theorem to the subsets C for

w w

suitably chosen sets X Table shows results for selected RLL constraints and co deword

lengths

Constructions of co des for channels with substitution asymmetric and bitshift errors as

well as b ounds on the maximum cardinalityofsuch co des of xed length have b een addressed

erreira and Lin FL Fredrick bynumerous other authors for example Blaum Blaum F

son and Wolf FW Immink Imm Kolesnik and Krachkovsky KolK Kuznetsov and

Vinck KuVa KuVb Lee and Wolf Lee LW LW Patap outian and Ku

mar PK Shamai and Zehavi SZ and Ytrehus Yta Ytb

CHAPTER ERRORCORRECTING CONSTRAINED CODING

q d k d k d k d k

Table Upp er b ounds on sizes of d k RLL constrained single shifterror correcting co des

of length q

CHAPTER ERRORCORRECTING CONSTRAINED CODING

Distance prop erties of sp ectralnull co des

Finallywe mention that sp ectralnull constrained co desin particular dcfree co deswith

Hamming errorcorrection capabilityhave received considerable attention See for example

Barg and Litsyn BL Blaum and van Tilb org TiBl Blaum Litsyn Buskens and

van Tilb org BLBT Calderbank Herro and Telang CHT Cohen and Litsyn CL

Etzion Etz Ferreira Fe Roth Roth Roth Siegel and Vardy RSV Waldman

and Nisenbaum WN Sp ectralnull co des also have inherent Hammingdistance prop er

ties as shown by Immink and Beenker ImmB They considered co des over the alphab et

g in which the orderm moment of every co deword x x x x vanishes for f

m K ie

i x m K

They referred to a co de with this prop erty as a co de with orderK zerodisparity For

each co deword x the discrete Fourier transform given by f x e where

satises

d f

for m K

This implies by part of Problem that the power sp ectral density of the ensemble of

sequences generated by randomly concatenating co dewords vanishes at f along with its

order derivatives for K A co de or more generally a constraint with this

prop erty is said to have an orderK spectralnul l at f

The following theorem from ImmB provides a lower bound on the minimum Ham

ming distance ofacode with sp ectral null at f

Theorem Let C be a code with orderK spectral nul l at f Let x y be distinct

codewords in C Then their Hamming distance satises

x y K

Hamming

This result will play a role in the subsequent discussion of co des for the Lee and Euclidean

metrics

Synchronizationbitshift error correction

Synchronization errors resulting from the insertion or deletion of symb ols and co ding meth

o ds for protection against such errors have b een the sub ject of numerous investigations The

CHAPTER ERRORCORRECTING CONSTRAINED CODING

edit distance intro duced by Levenshtein and often referred to as the Levenshtein metric is

particularly appropriate in this setting as it measures the minimum number of symbol in

sertions and deletions required to derive one nitelength sequence from another The reader

interested in co des based up on the Levenshtein metric is referred to Bours Bours Iizuka

Hasahara and NamahawaIKN Kruskal Krusk Levenshtein Lev Lev Lev

Lev Levenshtein and Vinck LV Tanaka and Kasai TK Tenengolts Ten

Ten and Ullman U U

When dealing with synchronization errors insertions and deletions of s in d k RLL

constrained systems it is convenien t to represent a constrained sequence as a sequence of

runs where a run corresp onds to a symbol along with the subsequent string of con

tiguous symbols preceding the next consecutive symbol We asso ciate to each run a

p ositiveinteger called the runlength representing the numb er of symb ols in the run As an

example the RLL sequence corresp onds to the sequence of runs

with runlengths

Let w bead k constrained sequence with n runs and corresp onding runlength sequence

s s s s Insertion of e symbols in the j th run of w generates the sequence with

runlengths s s s e s s while deletion of e symb ols from run j generates

j j n

the sequence of runlengths s s s e s s In the latter e cannot exceed

j j n

s An esynchronization error denotes such a pattern of e insertions or deletions o ccurring

within a single run Note also that a bitshift error or more generally an ebitshift error

consisting of e leftbitshift errors or e rightbitshift errors o ccurring at the b oundary b etween

two adjacent runs may be viewed as a pair of esynchronization errors in consecutive runs

one b eing an insertion error the other a deletion error

This runlengthoriented viewp oint has b een used in the design of RLL co des capable

of detecting and correcting bitshift and synchronization errors Hilden Howe and Wel

don Hild constructed a class of variable length co des named ShiftErrorCorrecting

Mo dulation SECM co des capable of correcting up to some presp ecied number of ran

dom ebitshift errors for a preselected shifterror size e The runlengths are regarded as

elements of a nite alphab et F whose size usually taken to be an odd prime integer satis

es k d jF j e The binary information string is viewed as a sequence of k

runs r r r r satisfying d k constraints with runlengths s s s s The

k k

sequence of transition p ositions t t t t is then dened by

t s mo d jF j for j k

j i

These values are then applied to a systematic enco der for an n k dBCH co de over a nite

eld F of prime size yielding parity symbols t t t A d k constrained binary

k k n

co deword is then generated by app ending to the original information runs the sequence of

parity runs r r r whose runlengths s s s satisfy

k k n k k n

d s djF j for j k kn j

CHAPTER ERRORCORRECTING CONSTRAINED CODING

and

t s mo d jF j for j k kn

j i

In particular for jF j t the resulting co de may be used to correct up to t random

bitshift errors where t is the designed errorcorrecting capability of the BCH co de Note

that a similar construction provides for correction of random esynchronization errors by

enco ding the runlengths themselves rather than the transition p ositions The interpretation

on runlengths of bitshift and more generally synchronization errors in terms of their eect

leads naturally to the consideration of another metric the Lee metric

The Lee distance x y of two symbols x y in a nite eld F of prime size is

Lee

the smallest absolute value of any integer congruent to the dierence x y mo dulo jF j

For vectors x y in F the Lee distance x y is the sum of the comp onentwise Lee

Lee

distances The Lee weight w x of a vector x is simply x where denotes the

Lee Lee

allzero vector of length n

Among the families of co des for the Leemetric are the wellknown negacyclic co des

intro duced by Berlekamp Berl Ch the family of cyclic co des devised by Chiang and

Wolf CW and the Leemetric BCH co des investigated by Roth and Siegel RS RS

All of these Leemetric co de constructions have the prop erty that the redundancy required

for correction of a Leemetric error vector of weight t is approximately t symb ols In contrast

co des designed for the Hamming metric require approximately t check symbols to correct

t random Hamming errors In a recording channel sub ject to esynchronization errors and

ebitshift errors where the predominant errors corresp ond to small values of e one might

anticipate reduced overhead using a Leemetric co ding solution This observation was made

indep endently by Roth and Siegel RS Saitoh Saia Saib and Bours Bours see

also Davydov Dav and KabatianskyDavydov and VinckKDV who have prop osed

avariety of constrained co de constructions based on the Leemetric and havederived b ounds

on the eciency of these constructions as we now describ e

For bitshifterror correction Saitoh prop osed a construction yielding co des with xed

is asymptotically optimal with binary symbol length He showed that the construction

resp ect to a Hamming b ound on the redundancy for singlebitshift errorcorrecting d k

RLL co des

The construction of Saitoh requires that the co dewords b egin with a symbol and end

with at least d symbols The co dewords will have a xed numb er of runs and consequently

a variable length in terms of binary symbols The co dewords are dened as follows If the

runlengths are denoted s i n the sequence of runlengths s for even values of i

i i

comprise a co deword in a singleerror correcting co de over the Lee metric The sequence of

runlengths s fori mo d comprise a co deword in a single errordetecting co de for the

Lee metric It is evident that in the presence of a single bitshift error the Leemetric single

errorcorrecting co de will ensure correct determination of the runlengths s for even values of i

CHAPTER ERRORCORRECTING CONSTRAINED CODING

i indicating if the erroneous runlength say s suered an insertion or deletion of a symbol

The Leemetric errordetecting co de will then complete the deco ding by determining if

the corresp onding deletion or insertion applies to runlength s or s

j j

In the broader context of synchronization errors Roth and Siegel describ ed and analyzed

a construction of d k RLL co des for detection and correction of such errors as an application

of a class of Leemetric BCH co des RS The shortened BCH co de of length n over a nite

prime eld F denoted C nr F is characterized by the paritycheck matrix

C B

C B H nr F

C B

r r

where is the locator vector consisting of distinct nonzero elements of the

smallest hdimensional extension eld F of F of size greater than n

Hence a word x x x x F is in C nr F if and only if it satises the

following r paritycheck equations over F

x for m r

The following theorem provides a lower b ound on the minimum Lee distance of C nr F

denoted d nr F

Lee

Theorem

r for r jF j

d nr F

Lee

jF j for jF j rjF j

This bound follows from Newtons identities ImmBKSa and can be regarded in

a way as the analogue of the BCH lower bound r on the minimum Hamming distance

of C nr F although the pro of of the r lower bound is slightly more complicated For

r jF j we can bound d nr F from belowby the minimum Hamming distance r

Lee

wer b ound do es not hold in general for all values of r however it do es hold for The r lo

all r in the baseeld case n jF j The r lower b ound for the baseeld case takes the

following form

Theorem For r n jF j

d nr F r Lee

CHAPTER ERRORCORRECTING CONSTRAINED CODING

The primitive case corresp onds to co des C nr F for which n jF j The redundancy

of such co des is known to be b ounded from ab ove by r h This b ound along with

the following lower b ound derived by a spherepacking argument combine to show that the

primitive co des are nearoptimal for suciently small values of r

Lemma Spherepacking bound Golomb and Welch GoW GoW A code

over a nite prime eld F of length n size M and minimum Lee distance r for some

r jF j must satisfy the inequality

n r

n i

jF j M

i i

Theorem A code over a nite prime eld F of length n size M and minimum

Lee distance r for some r jF j must satisfy the inequality

log n r log r n log M r

jF j jF j jF j

Pro of By Lemma we have

r n

jF j M

The theorem now follows by taking the logarithm to base jF j of b oth sides of this inequality

The construction of synchronizationerror correcting co des based up on the Leemetric

BCH co des is as follows Given constraints d k we cho ose jF j k d We re

gard every run of length s in the d k constrained information sequence as an element

sd mo d jF j of F and use a systematic enco der for C nr F to compute the cor

resp onding check symbols in F Each chec k symbol a in turn is asso ciated with a run of

length a d where a is the smallest nonnegative integer such that a a where

stands for the multiplicative unity in F The co de C nr F with r jF j and

n jF j can simultaneously correct b bitshift errors and s nonbitshift synchronization

errors whenever b s r Observe that when counting errors an ebitshift error is

counted as e bitshift errors this applies resp ectively also to synchronization errors Also

violate the d k constraint bitshift or synchronization errors may create runlengths that

In such a case we can mark the illegal runlength as an erasure rather than an error The

redundancy required will be no more than r h symbols from the alphab et F and

we recall that Theorem indicates the nearoptimalityof the Leemetric primitive BCH

h h

co des C jF j r F for values r jF j

Example Two typical choices for parameters d k are and b oth sat

isfying k d Setting jF j and r we obtain a family of co des for these

CHAPTER ERRORCORRECTING CONSTRAINED CODING

constraints based up on C n that can correct any error pattern of Lee weight and

detect error patterns of Lee weight In particular the co des will correct one singlebitshift

bitshift error or any other combination of two insertionsdeletions of symb ols For

n jF j the required redundancy is no more than h symbols

As mentioned ab ove the class of Hammingmetric SECM co des are directed primarily

toward the situation when only bitshifttyp e errors o ccur The constructions based up on Lee

metric co des can b e mo died to improve their eciency in this typ e of error environmentby

recording instead of the nominal co deword x x x x the dierentially preco ded

word y y y y dened by y x and y x x for i n where all

n i i i

op erations are taken mo dulo jF j If y is recorded and no bitshift errors occur the original

word x is reconstructed by an integration op eration

x y

i j

If however an ebitshift error occurs at the b oundary between runs j and j of y the

integration op eration converts the error into an esynchronization error in run j of x In other

words the original bitshift error pattern of Lee weighte is converted into a synchronization

error pattern of Lee weight e In order to ensure the correctness of the rst run y it suces

to require that the co de contain the allone word and all of its multiples

For the Leemetric BCH co des this construction provides the capability to correct up

to r bitshift errors and detect up to r bitshift errors when r jF j k d The

construction can b e extended to the baseeld case as well

Example Let jF j andr as in the previous example The construction ab ove

will generate co des with length n amultiple of For n the redundancy is r

runs for n the redundancy is r runs for n the

redundancy is r runs All of these co des will correct up to two singlebitshift

errors or one doublebitshift bitshift error By way of comparison in Hild Hilden et

al describ e SECM co des of lengths and for correcting two singlebitshift errors

requiring redundancy of and runs resp ectively These SECM co des do not handle

doublebitshift errors

Example As jF j increases so do es the discrepancy in the number of checksymbols

runs compared to the SECM co des in Hild For jF j suitable for representing

d k for example and r the Leemetric BCH co de with n requires check

symb ols for n the redundancy is symbols for n the

redundancy will be symb ols These co des will correct up to four singlebitshift errors

two singlebitshift and one doublebitshift errors or two doublebitshift errors The co des

presented in Hild for correcting up to four singlebitshift errors have lengths and

and require redundancy of and resp ectively

CHAPTER ERRORCORRECTING CONSTRAINED CODING

Bours Bours provided a construction of synchronizationerror correcting RLL co des

with xed length over the binary alphab et that also relies on an underlying Leemetric co de

He did not require the underlying co de to b e a Leemetric BCH co de however and thereby

avoided having the errorcorrection capability limited by the co de alphab et size

The denition of the Lee metric can also be generalized in a straightforward manner to

integer rings Orlitsky describ ed in Or a nonlinear construction of co des over the ring

of integers mo dulo for correcting any prescrib ed number of Lee errors His construction

is based on dividing a co deword of a binary BCH co de into nonoverlapping htuples and

regarding the latter as the Grayco de representations of the integers between and

It is also worth remarking that all of the Leemetric co des mentioned ab ove can be

eciently deco ded algebraically

We close the discussion of Leemetric co des by noting that the denition of the class of

Leemetric BCH co des was motivated by a Leemetric generalization of the result of Immink

and Beenker in Theorem to integervalued sp ectralnull constraints KSa EC

Theorem Let S be a constrained system over an integer alphabet with orderK

spectral nul l at f presented by a labeled graph G Let x y be distinct sequences in S

generatedbypaths in Gboth of which start at a common state u and end at a common state

v Then the Lee distance satises

x y K

Lee

This result will play an imp ortant role in the next section in the context of Euclidean

metric co des for PRML

When combining bitshift and synchronization errors any bitshift error can obviously b e

regarded as two consecutive synchronization errors in opp osite directions one einsertion

one edeletion thus reducing to the synchronizationonly mo del of errors However such

an approach is not optimal and b etter constructions have b een obtained to handle a limited

number of bitshift and synchronization errors combined See Ho d Ho d Klve Kl

and Kuznetsov and Vinck KuVa KuVb

Softdecision deco ding through Euclidean metric

Let x and y b e sequences of length n over the real numb ers The squar edEuclidean distance

between these sequences denoted x y isgiven by

Euclid

x y x y

i i

Euclid

CHAPTER ERRORCORRECTING CONSTRAINED CODING

The Euclidean metric is most relevant in channels with additive white Gaussian noise

AWGN In particular it is of interest in connection with the mo del of the magnetic record

ing channel as a binary input partialresp onse system with AWGN The success of trellis

co ded mo dulation as pioneered by Ungerb o eck in improving the reliabilityof memoryless

channels with AWGN provided the imp etus to design co ding schemes for channels with

memory such as partialresp onse channels in AWGN For binary inputrestricted partial

resp onse channels suitable as mo dels for recording channels such as the Class channel

characterized by the inputoutput relation y x x several approaches have been

i i i

prop osed that make use of binary convolutional co des These approaches typically require a

computer search of some kind to determine the co des that are optimal with resp ect to rate

Euclidean distance and maximumlikeliho o d detector complexity See for example Wolf

and Ungerb o eck WU Calderbank Heegard and Lee CHL Hole Hole and Hole

and Ytrehus HoY

There is another approach however that relies up on the concepts and co de construction

previous chapters The underlying idea is as techniques that have been develop ed in the

follows First nd a constrained system S presented by a lab eled graph G that ensures a

certain minimum Euclidean distance b etween the partialresp onse channel output sequences

generated when channel inputs are restricted to words in S Then apply statesplitting or

other metho ds to construct an ecient enco der from binary sequences to S Since the graph

structure E underlying the enco der is often more complex in terms of numb er of states and

interconnections than the original graph G use G rather than E as the starting point for

the trellisbased Viterbi detector of the co ded channel

Karab ed and Siegel KSa showed that this approach can be applied to the family of

constrained systems S whose sp ectral null frequencies coincide with those of the partial

resp onse channel transfer function The resulting co des are referred to as matchedspectral

nul l codes We conclude this section with a brief summary of the results that p ertain to the

application of this technique to the Class and related partialresp onse systems We will

refer to the dicode channel whichischaracterized by the inputoutput relation y x x

i i i

and has a rstorder sp ectral null at f andwe remark that the Class partialresp onse

channel maybeinterpreted as a pair of interleaved dico de channels one op erating on inputs

with even indices the other on inputs with o dd indices

ea constrained system over an integer alphabet with orderK spec Lemma Let S b

tral nul l at zero frequency Let S be the constrained system of sequences at the output of

a cascade of N dicode channels with inputs restricted to words in S Then S has an

orderK N spectral nul l at zero frequency

Noting that any lower b ound on Lee distance provides a lower bound on squared

Euclidean distance for sequences over integer alphab ets we obtain from the preceding lemma

and Theorem the following lower b ound on the minimum squaredEuclidean distance of

a binary matchedsp ectralnull co ded partialresp onse channel with sp ectral null at f

CHAPTER ERRORCORRECTING CONSTRAINED CODING

Theorem Let S beaconstrained system over the alphabet f g with orderK

spectral nul l at zero frequency Let x and y be distinct sequences in S diering in a nite

number of positions If x and y are the corresponding output sequences of a partialresponse

channel consisting of a cascade of N dicode channels then

K N x y

Euclid

It is easy to see that the lower bound of Theorem remains valid in the presence

of J way symb olwise interleaving of the constrained sequences and the partialresp onse

channel In particular for the Class partialresp onse channel ie the way interleaved

dico de channel the application of sequences obtained bywayinterleaving a co de having a

rstorder sp ectral null at zero frequency doubles the minimum squaredEuclidean distance

at the channel output relative to the unco ded system

We remark that J way interlea ving of sequences with an orderK sp ectral null at f

generates sequences with orderK sp ectral nulls at frequencies f rJ forr J

MS Thus a way interleaved dcfree co de has sp ectral nulls at zero frequency and at

frequency f corresp onding to the sp ectral null frequencies of the Class partial

resp onse channel

Graph presentations for sp ectral null sequences are provided by canonical diagrams in

tro duced by Marcus and Siegel MS for rstorder sp ectral null constrain ts and then

extended to highorder constraints by among others Monti and Pierob on MPi Karab ed

and Siegel KSa Eleftheriou and Cideciyan EC and Kamab e Kam

Discussion of the canonical diagrams requires the notion of a lab eled graph with an innite

number of states Sp ecically a countablestate labeled graph G V E L consists of a

countablyinnite set of states V a set of edges E where eachedge has an initial state and

a terminal state b oth in V and the states in V have b ounded outdegree and indegree and

an edge lab eling L E where is a nite alphab et

We say that a countablestate graph G is a periodp canonical diagram for a sp ectral

null constraint if

Every nite subgraph H G generates a set of sequences with the prescrib ed sp ectral

null constraint

For any p erio dp graph G that presents a system with the sp ecied sp ectral null

constraint there is a lab elpreserving graph homomorphism of G into G meaning

a map from the edges of G to the edges of G that preserves initial states terminal

states and lab els

The canonical diagram G for a rstorder sp ectral null constraint at zero frequency

with symbol alphab et f g is shown in Figure As mentioned in Example

CHAPTER ERRORCORRECTING CONSTRAINED CODING

Figure Canonical diagram for rstorder sp ectral null at f

the capacity of the constrained system generated by a subgraph G consisting of B

consecutive states and the edges with b eginning and ending states among these is given by

capS G log cos

From this expression it follows that

lim capS G

We p ointed out in Chapter that the constrained system generated byany nite subgraph

of G is almostnitetyp e Applying Theorem we see that bycho osing B large enough

we can construct a slidingblo ck deco dable nitestate enco der for the constrained system

S G at any presp ecied rate p q with pq

From the structure of the canonical diagram it is clear that any constrained system with

rstorder sp ectral null at f limits the numb er of consecutive zero samples at the output

of the dico de channel When the constrained sequences are twiceinterleaved and applied to

the Class partialresp onse channel the number of zero samples at the output is limited

globally as well as in each of the eveno dd interleaved subsequences This condition is

analogous to that achieved by the GI constraints for the baseline PRML system

The subgraph G chosen for the co de construction may be augmented to incorp orate

the dico de channel memory as shown in Figure for the case M providing the

basis for a dynamic programming Viterbi detection algorithm for the co deddico de channel

with AWGN Each state in the trellis has a lab el of the form v where v is the state in

Figure from which it is derived and the sup erscript m denotes the sign of the dico de

channel memory Just as do es the unco ded dico de detector graph represented by the trellis

in Figure of Chapter the co deddico de detector graph supp orts sequences that can

cause p otentially unbounded delays in the merging of survivor sequences and therefore in

deco ding The sp ectralnull co de sequences that generate these output sequences are called

quasicatastrophic sequences and they are characterized in the following prop osition

esented Prop osition The quasicatastrophic sequences in the constrained system pr

by G are those generated by more than one path in G

B B

To limit the merging delay the matchedsp ectralnull co de is designed to avoid these

sequences and it is shown in KSa that this is always p ossible without incurring a rate

loss for any G with B B

CHAPTER ERRORCORRECTING CONSTRAINED CODING

Figure Graph underlying co deddico de channel Viterbi detector for G

Further details and developments regarding the design and application of matched

sp ectralnull co des to PRML systems may be found in Shung Thap Thap

Fred and Rae

Forbidden list co des for targeted error events

This section whichisyet to b e written will b e based on results taken from Karab edSiegel

Soljanin KSS

Problems

Problem AgraphG is called binary if its lab els are over the alphab et f g The Hamming

weight of a word w generated by a binary graph G is the number of s in w The weight of a path

in a binary graph G is the weightoftheword generated by that path

CHAPTER ERRORCORRECTING CONSTRAINED CODING

Given a binary graph G and states u and v in Gdenoteby the numb er of paths of length

uv k

and weight k in G that originate in u and terminate in v For states u and v in G dene the

length weightdistribution polynomial of paths from u to v in the indeterminate z by

P z z

uv k

As an example for the graph H in Figure P z z since there are three paths of

length that originate in A and terminate in C one path has weight and the other paths each

has weight

A C

Figure Graph H for Problem

For a binary graph G V E L let B z be the jV j jV j matrix in the indeterminate z

where

z B z P

G uv

for every u v V Eachentry in B z is therefore a p olynomial in z of degree at most

Compute B z for the graph in Figure

For the matrix B z found in compute B z and B z

H H H

Let B z b e the matrix asso ciated with a binary graph Gandlet u and v b e states in G

Given a p ositiveinteger obtain an expression for the p olynomial P z in terms of B z

Identify the matrix B asso ciated with a binary graph G

Identify the matrix B

Let G b e a binary graph and let z b e a p ositiverealnumber Show that G is irreducible if

and only if the matrix B z is irreducible Do es this hold also when z

Problem Recall the denitions from Problem Let G b e a binary lossless graph and let u

and v b e states in G For p ositiveintegers and ddenoteby J the number of w ords of length

uv d

and weight d that can b e generated in G by paths that originate in u and terminate in v

CHAPTER ERRORCORRECTING CONSTRAINED CODING

Show that for every d

uv k uv d

Based on show that for every real z in the range z

k d

J z

uv d uv k

and therefore

J min z P z

uv d

Based on and derive an upp er b ound on the number of words of length and weight d

in S G as a function of B z andd

Problem Recall the denitions from Problem Let S be a constrained system presented

by a deterministic binary graph G with nite memory M For nonnegativeintegers and k denote

by Y the numb er of ordered pairs w w ofwords of length in S such that w and w are at

Hamming distance k ie they dier on exactly k lo cations Dene the p olynomial Y z by

Y z Y z

Show that Y jS f g j

Show that Y Y

Let S denote the RLL constrained system ShowthatwhenS S the p olynomial

Y z can b e written as

Y z zz B z

where is the allone column vector and

z z

C B

B z

C B

Compute Y z explicitly for S S

Find the largest integer n for which there exists a blo ck S nenco der whose co dewords

are at Hamming distance at least from each other

Generalize for any constrained system S over f g with nite memory M